Created by T. Madas

Created by T. Madas

Question 13 (***)

The curve C has equation

2cos3 sin 1x y = , 0 ,x y π≤ ≤ .

a) Show that

3tan 3 tandy

x ydx

= .

The point ,12 4

Pπ π

lies on C .

b) Show that an equation of the tangent to C at P is

3y x= .

proof

Created by T. Madas

Created by T. Madas

Question 14 (***)

A curve has equation

2 23 2 4 1x xy y x y− + + − = .

a) Show clearly that

2 6

4 2

dy x y

dx y x

+ −=

− +.

b) Hence show further that the value of x at the stationary points of the curve satisfies the equation

2 5

33x = .

proof

Created by T. Madas

Created by T. Madas

Question 39 (****)

The equation of a curve is given implicitly by

2 3 34 e xy y x C+ = + ,

where C is a non zero constant.

a) Find a simplified expression for dy

dx.

The point ( )1,P k , where 0k > , is a stationary point of the curve.

b) Find an exact value for C .

C4L , ( )

( )

2 2 3

3

3 e

2 2 e

x

x

x ydy

dx y

−=

+,

324eC

−=

Created by T. Madas

Created by T. Madas

Question 40 (****)

A curve C has implicit equation

2 1

3

xy

xy

+=

+.

a) Find an expression for dy

dx, in terms of x and y .

b) Show that there is no point on C , where the tangent is parallel to the y axis.

C4N , 22

2 3

dy y

dx xy

−=

+

Created by T. Madas

Created by T. Madas

Question 49 (****)

The equation of a curve is given implicitly by

2 2 1y x− = , 1y ≥ .

Show clearly that

2 2 2

2 3

d y y x

dx y

−= .

proof

Created by T. Madas

Created by T. Madas

Question 4 (**+)

The binomial expression ( )131 x+ is to be expanded as an infinite convergent series, in

ascending powers of x .

a) Determine the expansion of ( )131 x+ , up and including the term in 3x .

b) Use the expansion of part (a) to find the expansion of ( )131 3x− , up and

including the term in 3x .

c) Use the expansion of part (a) to find the expansion of ( )1327 27x− , up and

including the term in 3x .

( )2 3 451 11 3 9 81x x x O x+ − + + , ( )2 3 451 3x x x O x− − − + ,

( )2 3 4513 3 27x x x O x− − − +

Created by T. Madas

Created by T. Madas

Question 5 (**+)

( )( )( )

5 3

1 1 3

xf x

x x

+=

− +,

1

3x < .

a) Express ( )f x into partial fractions.

b) Hence find the series expansion of ( )f x , up and including the term in 3x .

C4B , ( )2 1

1 1 3f x

x x= +

− +, ( ) ( )2 3 43 11 25f x x x x O x= − + − +

Question 6 (**+)

( )( )

3

2

1 2

xf x

x=

+, 12x ≠ − .

a) Find the first 4 terms in the series expansion of ( )f x .

b) State the range of values of x for which the expansion of ( )f x is valid.

C4B , ( ) ( )2 3 4 52 12 48 160f x x x x x O x= − + − + , 1 12 2x− < <

Created by T. Madas

Created by T. Madas

Question 11 (***)

4 12y x= − , 1 1

3 3x− < < .

a) Find the binomial expansion of y in ascending powers of x up and including

the term in 3x , writing all coefficients in their simplest form.

b) Hence find the coefficient of 2x in the expansion of

( )( )1212 4 4 12x x− − .

C4M , ( )2 3 49 272 3 4 8y x x x O x= − − − + , 27−

Created by T. Madas

Created by T. Madas

Question 12 (***)

The binomial ( )121 x

−+ is to be expanded as an infinite convergent series, in ascending

powers of x .

a) Find the series expansion of ( )121 x

−+ up and including the term in 3x .

b) Use the expansion of part (a) to find the expansion of 1

1 2x+, up and

including the term in 3x .

c) State the range of values of x for which the expansion of 1

1 2x+ is valid.

d) Use the expansion of 1

1 2x+ with 0.1x = − to show that 5 2.235≈ .

( )2 3 43 511 2 8 16x x x O x− + − + , ( )2 3 43 51 2 2x x x O x− + − + ,

1 12 2x− < <

Created by T. Madas

Created by T. Madas

Question 13 (***)

( ) 1 2f x x= − , 1

2x < .

a) Expand ( )f x as an infinite series, up and including the term in 3x .

b) By substituting 0.01x = in the expansion, show that 2 1.414214≈ .

( ) ( )2 3 41 11 2 2f x x x x O x= − − − +

Created by T. Madas

Created by T. Madas

Question 4 (**+)

The figure above shows the curve C , given parametrically by

3 3x t= + , 2

3y

t= , 0t > .

The finite region R is bounded by C , the x axis and the straight lines with equations 4x = and 11x = .

a) Show that the area of R is 3 square units.

The region R is revolved in the x axis by 2π radians to form a solid of revolution S .

b) Find the volume of S .

4

3V

π=

x

y

4

C

R

O 11

Created by T. Madas

Created by T. Madas

Question 5 (***)

The figure above shows the curve C , given parametrically by

2 26 ,x t y t t= = − , 0t ≥ .

The curve meets the x axis at the origin O and at the point P .

a) Show that the x coordinate of P is 6 .

The finite region R , bounded by C and the x axis, is revolved in the x axis by 2π radians to form a solid of revolution, whose volume is denoted by V .

b) Show clearly that

( )22

0

12T

V t t t dtπ= − ,

stating the value of T .

c) Hence find the value of V .

C4J , 1T = , 5

Vπ

=

x

y

P

C

R

O

Created by T. Madas

Created by T. Madas

Question 11 (***+)

The figure above shows a curve known as a re-entrant cycloid, with parametric equations

4sin , 1 2cosx yθ θ θ= − = − , 0 2θ π≤ ≤ .

The curve crosses the x axis at the points P and Q .

a) Find the value of θ at the points P and Q .

b) Show that the area of the finite region bounded by the curve and the x axis, shown shaded in the figure above, is given by the integral

2

1

21 6cos 8cos dθ

θ

θ θ θ− + ,

where 1θ and 2θ must be stated.

c) Find an exact value for the above integral.

5,

3 3

π πθ = , 1 2

5,

3 3

π πθ θ= = ,

204 3

3

π+

x

y

P QO

Created by T. Madas

Created by T. Madas

Question 12 (***+)

The figure above shows the curve C , with parametric equations

2 , 1 cosx t y t= = + , 0 2t π≤ ≤ .

The curve meets the coordinate axes at the points A and B .

a) Show that the area of the shaded region bounded by C and the coordinate axes is given by the integral

( )2

1

2 1 cost

t

t t dt+ ,

where 1t and 2t are constants to be stated.

b) Evaluate the above parametric integral to find an exact value for the area of the shaded region.

1 20,t t π= = , 2area 4π= −

x

y

O

A

B

C

Created by T. Madas

Created by T. Madas

Question 16 (****)

The figure above shows the curve C , with parametric equations

4cos , sinx yθ θ= = , 02

πθ≤ ≤ .

The curve meets the coordinate axes at the points A and B . The straight line with

equation 12y = meets C at the point P .

a) Show that the area under the arc of the curve between A and P , and the x axis, is given by the integral

2 2

6

4sin d

π

πθ θ .

The shaded region R is bounded by C , the straight line with equation 12y = and the

y axis.

b) Find an exact value for the area of R .

( )1area 4 3 36 π= −

x

y

O

A

B

PR

12y =

Created by T. Madas

Created by T. Madas

Question 12 (***)

A curve C is given parametrically by

4 1x t= − , 5

102

yt

= + , t ∈ , 0t ≠ .

The curve C crosses the x axis at the point A .

a) Find the coordinates of A .

b) Show that an equation of the tangent to C at A is

10 20 0x y+ + = .

c) Determine a Cartesian equation for C .

( )2,0− , ( )( )( )10 2

1 10 10 or 1

xx y y

x

++ − = =

+

Created by T. Madas

Created by T. Madas

Question 13 (***)

A curve C is given parametrically by

3 1x t= − , 1

yt

= , t ∈ , 0t ≠ .

Show that an equation of the normal to C at the point where C crosses the y axis is

13

3y x= + .

proof

Created by T. Madas

Created by T. Madas

Question 24 (***+)

A curve C is defined by the parametric equations

cos , cos 2x t y t= = , 0 t π≤ ≤ .

a) Find dy

dx in its simplest form.

b) Find a Cartesian equation for C .

c) Sketch the graph of C .

The sketch must include

•••• the coordinates of the endpoints of the graph.

•••• the coordinates of any points where the graph meets the coordinates axes.

4cosdy

tdx

= , 22 1y x= − , ( )( ) ( ) ( )( )2 21,1 1,1 , 0, 1 , ,0 ,02 2− − −

Created by T. Madas

Created by T. Madas

Question 25 (***+)

A curve C is given by the parametric equations

3 2

1

tx

t

−=

−,

2 2 2

1

t ty

t

− +=

−, t ∈ , 1t ≠ .

a) Show clearly that

22dy

t tdx

= − .

The point ( )51, 2P − lies on C .

b) Show that the equation of the tangent to C at the point P is

3 4 13 0x y− − = .

proof

Created by T. Madas

Created by T. Madas

Question 49 (****)

A curve C is given by the parametric equations

( )sec , ln 1 cos 2 , 02

x yπ

θ θ θ= = + ≤ < .

a) Show clearly that

2cosdy

dxθ= − .

The straight line L is a tangent to C at the point where 3

πθ = .

b) Find an equation for L , giving the answer in the form y x k+ = , where k is an exact constant to be found.

c) Show that a Cartesian equation of C is

2 e 2yx = .

2 ln 2y x+ = −

Created by T. Madas

Created by T. Madas

Question 3

Carry out the following integrations:

1. 4 4 4121 1

e e e8 32

x x xx dx x C= − +

2. 5 5

5 sin 4 cos 4 sin 44 16

x x dx x x x C= − + +

3. ( ) ( )1 1

2 1 cos 2 2 1 sin 2 cos 22 2

x x dx x x x C+ = + + +

4. 3 3

3 cos4 sin 4 cos44 16

x x dx x x x C− = − − +

5. 2 2 2 2 2 21 1 1

e e e e2 2 4

x x x xx dx x x C− − − −= − − − +

6. 2 21 2 2

sin 5 cos5 sin 5 cos55 25 125

x x dx x x x x x C= − + + +

7. 2 21 1 1 13 3 3 3cos 3 sin 18 cos 54sinx x dx x x x x x C= + − +

8. 3 4 4121 1

ln ln8 32

x x dx x x x C= − +

9. 2 21 1

ln 3 ln32 4

x x dx x x x C= − +

10. 3 2 2

ln ln 1

2 4

x xdx C

x x x= − − +

Created by T. Madas

Created by T. Madas

Created by T. Madas

Created by T. Madas

Question 6

( ) 9sin 12cosf x x x≡ + , x ∈ .

a) Express ( )f x in the form ( )sinR x α+ , 0R > , 02

πα< < .

b) Hence, solve the trigonometric equation

9sin 12cos 7.5x x+ = , 0 2x π< < .

( ) ( )c9sin 12cos 15sin 0.927f x x x x≡ + ≅ + , c c1.69 , 5.88x ≈

Created by T. Madas

Created by T. Madas

Question 9

( ) 4sin 3cosf θ θ θ≡ + , θ ∈ .

a) Write the above expression in the form ( )sinR θ α+ , 0R > , 0 90α< < ° .

b) Write down the maximum value of ( )f θ .

c) Find the smallest positive value of θ for which this maximum value occurs.

( ) ( )4sin 3cos 5sin 36.9f θ θ θ θ≡ + ≅ + ° , ( )max 5f θ = , 53.1θ ≈ °

Created by T. Madas

Created by T. Madas

Question 10

( ) sin 3 cosf x x x≡ − , x ∈ .

a) Express ( )f x in the form ( )sinR x α− , 0R > , 02

πα< < .

b) Write down the maximum value of ( )f x .

c) Find the smallest positive value of x for which this maximum value occurs.

( ) sin 3 cos 2sin3

f x x x xπ

≡ − ≡ − , ( )max 2f x = , 5

6x

π=

Created by T. Madas

Created by T. Madas

Question 15

( ) 3sin cosf x x x≡ + , x ∈

a) Express ( )f x in the form ( )cosR x α− , 0R > , 02

πα< < .

b) Solve the equation

( ) 2f x = for 0 2x π< < .

c) Write down the minimum value of ( )f x .

d) Find the smallest positive value of x for which this minimum value occurs.

( ) ( )c10 cos 1.249f x x≅ − , c c0.363 ,2.135x = , ( )min 10f x = − , c4.391x =

Created by T. Madas

Created by T. Madas

Question 17

( ) 2sin 2cosf x x x≡ + , x ∈ .

a) Express ( )f x in the form ( )sinR x α+ , 0R > , 02

πα< < .

b) State the minimum and the maximum value of …

i. … 2

y f xπ

= − .

ii. … ( )2 1y f x= + .

iii. … ( )2

y f x= .

iv. … ( )

10

3 2y

f x=

+.

( ) 8 sin4

f x xπ

≡ + , 8, 8− , 2 8 1,2 8 1− + + , [ ]0,8 , 2,5 2

Created by T. Madas

Created by T. Madas

Question 7 (***)

4sin 7cosx θ θ= + .

The value of θ is increasing at the constant rate of 0.5 , in suitable units.

Find the rate at which x is changing, when 2

πθ = .

C4I , 72−

Question 8 (***)

Fine sand is dropping on a horizontal floor at the constant rate of 4 3 1cm s− and forms

a pile whose volume, V 3cm , and height, h cm , are connected by the formula

48 64V h= − + + .

Find the rate at which the height of the pile is increasing, when the height of the pile has reached 2 cm .

C4M , 15 2.24 cms−≈

Created by T. Madas

Created by T. Madas

Question 9 (***)

An oil spillage on the surface of the sea remains circular at all times.

The radius of the spillage, r km , is increasing at the constant rate of 10.5 km h− .

a) Find the rate at which the area of the spillage, A 2km , is increasing, when the circle’s radius has reached 10 km .

A different oil spillage on the surface of the sea also remains circular at all times.

The area of this spillage, A 2km , is increasing at the rate of 0.5 2 1km h− .

b) Show that when the area of the spillage has reached 210 km , the rate at which the radius r of the spillage is increasing is

1

4 10π1km h− .

2 110 31.4 km hπ −≈

Created by T. Madas

Created by T. Madas

Question 10 (***)

Liquid dye is poured onto a large flat cloth and forms a circular stain, the area of

which grows at a steady rate of 1.5 2 1cm s− .

Calculate, correct to three significant figures, …

a) … the radius, in cm , of the stain 4 seconds after it started forming.

b) … the rate, in 1cms− , of increase of the radius of the stain after 4 seconds.

C4C , 6

1.38 cmrπ

= ≈ , 13

0.173 cms32π

−≈

Question 11 (***)

The variables y , x and t are related by the equations

( )3

2715 4

3y

x= −

+ and ( )

1ln 3

3x t+ = , 3x > − .

Find the value of dy

dt, when 9x = .

C4K , 15

64

dy

dt=

Created by T. Madas

Created by T. Madas

Question 12 (***+)

Two variables x and y are related by

( )21 44y x xπ= − .

The variable y is changing with time t , at the constant rate of 0.2 , in suitable units.

Find the rate at which x is changing with respect to t , when 2x = .

10.0637

5π≈

Question 13 (***+)

The variables y , x and t are related by the equations

15 110e

xy

−= and 6 1x t= + , 0t ≥ .

Find the value of dy

dt, when 4t = .

C4N , 4

6

5t

dy

dt ==

Created by T. Madas

Created by T. Madas

Question 14 (****)

Liquid is pouring into a container at the constant rate of 30 3 1cm s− .

The container is initially empty and when the height of the liquid in the container is

h cm the volume of the liquid, V 3cm , is given by

236V h= .

a) Find the rate at which the height of the liquid in the container is rising when the height of the liquid reaches 3 cm .

b) Determine the rate at which the height of the liquid in the container is rising 12.5 minutes after the liquid started pouring in.

C4O , 15 0.139 cms36−

= , 11 0.0167 cms60−

=

ImplicitInfinite BinomialParametric IntegrationParametricsPartsR MethodRates of Change